Coherence protection of spin qubits in hexagonal boron nitride

Spin defects in foils of hexagonal boron nitride are an attractive platform for magnetic field imaging, since the probe can be placed in close proximity to the target. However, as a III-V material the electron spin coherence is limited by the nuclear spin environment, with spin echo coherence times of ∽100 ns at room temperature accessible magnetic fields. We use a strong continuous microwave drive with a modulation in order to stabilize a Rabi oscillation, extending the coherence time up to ∽ 4μs, which is close to the 10 μs electron spin lifetime in our sample. We then define a protected qubit basis, and show full control of the protected qubit. The coherence times of a superposition of the protected qubit can be as high as 0.8 μs. This work establishes that boron vacancies in hexagonal boron nitride can have electron spin coherence times that are competitive with typical nitrogen vacancy centres in small nanodiamonds under ambient conditions.

The Hamiltonian of the S = 1 crystal ground-state can be expressed as H e .
is composed of a zero-field splitting with D = 3.47 GHz, and E ≈ 59 MHz, see Supplementary  Fig. 2, and an electron Zeeman term with gyromagnetic ratio γ e ≈ 28 MHz/mT. S j are the S=1 electron spin operators. The D-term defines a quantisation-axis along the c-axis of the crystal, and the static B-field is applied along this direction. To control the spin, an alternating magnetic-field is applied along the x-axis, and has the form H c (t).
H c (t) = [Ω cos (ωt + ϕ) + 2ϵ m sin (ωt + ϕ) sin (ω m t − θ m )]S x (2) The first term is the usual Rabi drive. The second term adds an amplitude modulated field in quadrature with the Rabi drive. To simplify the discussion we consider only a two-level system comprised of the m s = 0, m s = +1 states that are near resonance with the Rabi drive, i.e. S → σ, H e → 1 2 ω 0 σ z . Transforming to the first rotating frame of the Rabi drive: H ′ c = e iωtσz/2 H c e −iωtσz/2 . In the case ϕ = 0, where σ ′ α are the Pauli spin-1/2 matrices, and the superscript identifies the frame. The counterrotating term has been neglected. The Hamiltonian can be expressed as an effective magnetic field Ω ′ ef f (t) that causes the spin-1/2 to rotate. We consider the resonant case, where the detuning ∆ = ω 0 − ω = 0. If ϵ m = 0, the Ω-term drives an unprotected Rabi oscillation about the x ′ -axis. In this frame, the ϵ m term provides an AC B-field, that causes the direction of the effective magnetic field to rock in the x ′ y ′ -plane, as will be discussed later.
Making a second rotating frame approximation to rotate into the dressed states basis, In the second rotating frame, the control field is equivalent to a DC magnetic field Ω ′′ ef f (t) of magnitude ϵ m that is oriented using the phases of the drive θ m and ϕ. By setting the drive such that Ω = ω m , θ m = ϕ = 0, the Hamiltonian reduces to H ′′ C = 1 2 ϵ m σ ′′ z , and we can define a protected basis. In the following sections, we will present a qualitative discussion of how this control field can protect against decoherence.
Supplementary Note 2: Locking of Rabi frequency to modulation frequency, ω m We start by considering the damping of an unprotected single-drive Rabi oscillation (ϵ m = 0) due to low frequency fluctuations in the Rabi frequency, most likely caused by errors in the microwave power applied. To this end we numerically calculate the Bloch-vector dynamics in the first rotating frame with no damping using the Heisenberg equationσ , which can be expressed as a We consider the effect of a DC error by randomly picking a Rabi frequency from a Gaussian distribution of standard deviation 0.1Ω. The results are shown in the left-hand movie of Supplementary Movie 1. The blue arrow shows the rotation of the Bloch-vector with no Rabi frequency error. The grey vectors show that a spread in Rabi frequencies causes a divergence in the rotation angle of the Rabi oscillation. Consequently, the time ensemble averaged Rabi oscillation is damped, and this is seen as a shrinking of the ensemble averaged Bloch-vector (green). The Rabi oscillation is less sensitive to errors in the detuning, since the effective Rabi frequency depends quadratically on the detuning for Ω ≫ δ, since Ω ef f = √ Ω 2 + δ 2 . A similar calculation is made for the case of the CCD drive (ϵ m ̸ = 0), and is shown as the right-hand movie in Supplementary Movie 1. In the CCD case, the effective magnetic field (Black) performs a rocking motion in the x ′ y ′ -plane about the x ′ -axis, at a frequency ω m . This motion limits the spread in the Bloch-vectors (grey), and a clear refocusing behaviour can be seen at a frequency ϵ m . This locks the Rabi frequency to the modulation frequency ω m and results in a suppression of the damping of the time-ensemble averaged Bloch vector (green). Further insight can be obtained by considering the dressed basis, where the Hamiltonian is given by H ′′ C . In this frame, the rotating basis is such that σ ′′ = (0, 0, −1) = |0 ′′ ⟩⟨0 ′′ | corresponds to a Rabi oscillation with frequency ω m that starts at time t=0, and σ In the case of an unprotected Rabi oscillation (ϵ m = 0), for a Rabi-drive that is matched to the clock, (Ω = ω m ) there is no effective magnetic field, and the Bloch-vector is stationary. However, an error in the Rabi frequency generates an effective field term 1 x , causing the Bloch-vector to rotate about x ′ . As before, a spread in the Rabi frequency leads to a divergence in the rotation angle of the Bloch-vector and a damping of the time ensemble average of the Bloch-vector. When θ m = 0, the ϵ m -component gives rise to an effective magnetic field along z ′′ . Now, the effect of an error in the Rabi frequency with respect to the clock ω m is to tilt the effective field away from z ′′ . This limits the rotation about x ′ , and limits the admixing of the Rabi oscillation |0 ′′ ⟩ with |1 ′′ ⟩.
In the lab frame, this results in a small reduction in the contrast of the Rabi oscillation that does not increase with time. The signal will acquire sidebands at the frequencies ± ϵ 2 m + (Ω − ω m ) 2 .
Supplementary Figure 1: In the protected frame, the ϵ m -term limits the rotation of the Blochvector due to an error in the Rabi frequency. A view of the x"z"-plane in the dressed states basis is presented. In the lab frame, the state (0,0,-1) is observed as a Rabi oscillation that is in phase with a clock that starts at time t=0 at a frequency ω m , see Eq. 17. The state (0,0,1) has a π-phase shift compared to (0,0,-1). In the protected frame, a detuning of the Rabi frequency from the clock frequency ω m causes a rotation about x ′′ . Left unchecked, this will cause a divergence in the phase of the Rabi oscillation, and damping of the Rabi oscillation. Adding an ϵ m -drive with θ m = ϕ = 0 applies an effective field along the z ′′ -direction. Now an error in the Rabi frequency with respect to the modulation frequency, Ω − ω m tilts the effective magnetic field. This results in a precession of the Bloch-vector about the effective magnetic field, limiting the spread in the Bloch-vector.

Supplementary Note 3: Continuous dynamic decoupling of qubit from noise
In this section, we analyze how by using the CCD drive to define a rotating qubit basis, the interactions of the qubit with a fluctuating magnetic field can be shifted in frequency. This can be used to enhance coherence times, by moving to a lower noise frequency regime. For simplicity, consider a two-level system in a static B-field along z with a Hamiltonian (h = 1) In addition to the DC B-field, the spin is subject to a fluctuating B-field with a Hamiltonian of the form, To calculate the decoherence rates, we use the Bloch-Redfield equation to calculate changes to the density matrix ρ ′ in the stationary frame of the control Hamiltonian H c (t), i.e. ρ ′ = e iHct ρe −iHct .
Making the Born-Markov approximation ρ ′ (t−τ ) → ρ ′ (t), and the upper limit on the integral t → ∞ gives G α (t) describes the correlations in the fluctuating B-field components, which are assumed independent of each other and local time t. The noise power spectrum is described by F α (ω). In a basis that is stationary with respect to the control Hamiltonian, H c , the components of the fluctuating magnetic field are transformed by a rotation at a frequency ω = ω 0 , such that where ϵ αβγ is the Levi-Cevita symbol. This leads to a sampling of the noise spectrum at a frequency ω, which determines the decoherence rates.
For the case of a freely evolving spin, H c = H 0 = 1 2 ω 0 σ z . The z-fluctuations, leads to phase-flip errors and a dephasing rate proportional to the DC noise.
The x-fluctuations, give rise to bit-flip errors. The rotation with frequency components ±ω 0 samples the noise spectrum giving rise to contributions from F (±ω). This can be viewed as the energy-cost of a bit-flip.
In general, the noise spectrum is stronger at low frequency, leading to a fast damping of the Ramsey interference that is dominated by Γ * 2 = 4F (0). Another key point is that the energy-gap induced by the static B z -field only protects against bit-flip errors. Phase-flip errors are prone to low frequency fluctuations in the size of the energy gap, due to fluctuations in B z . Now consider the case of a qubit under a continuous drive. To evaluate the frequency components of the noise spectrum responsible for the damping of the qubit state, we perform a series of rotations on the (x,y,z)-components of the fluctuating magnetic field. Each rotation arises from a term in the control Hamiltonian, and are carried out using the largest energy term first. The results are summarized in Supplementary Table 1. For example, a rotation about the z-axis, leaves the zcomponent unchanged, whilst mixing the frequencies present in the xy spin-components.

Frame
Rotation x,y z Free Table 1: Frequency components of the noise spectrum responsible for damping under free-evolution, single and double drive conditions (θ m = 0). Under each rotation about the σ γ axis, with frequency ω γ the γ spin-component is unchanged. The other spin-components are shifted in frequency by ±ω γ , and shared between the two spin-components that are not γ. Each entry gives a vector, with the frequency-shifts of each spin component. x indicates that no spin-component is present. & indicates that both frequencies are present. For every frequency presented, the negative value is also present.
In a Ramsey interference experiment, the qubit freely evolves, and the main contributions to damping are F z (0) and F x,y (ω 0 ). Usually, F z (0) is worse due to low frequency noise.
In a single drive Rabi experiment, the main contributions to damping are F x (ω 0 ), F x (ω 0 ± Ω), and F z (Ω). F z (Ω) dominates in the low Ω regime. However, as the Rabi drive increases, the power fluctuations in the drive field lead to an increasing noise at F x (ω 0 ). This can be seen in the experiments, see Fig. 1f in main paper.
In the CCD experiment, the main x-noise terms are F x (ω 0 ± ϵ m ),F x (ω 0 ± ω m ), and F x (ω 0 ± ω m ± ϵ m ) are now shifted away from the drive frequency. The z-noise terms F z (ω m ) and F z (ω m ± ϵ m ) are shifted away from DC, and provided |ω m − ϵ m | ≫ cut-off frequency of the noise spectrum, the decoherence should be reduced. For further discussion of the protection against noise see Wang et al. [ This is a sum of the linear susceptibilities of the two ESR transitions from m s = 0 → ±, χ(x) = γ/(γ + ix).

Supplementary Note 5: T 1 decay in CCD measurements
In the CCD experiment shown in Fig. 2 of the main text, and similarly in Supplementary Figure  4b, there is a decay of the stabilized Rabi oscillation, but also a decay of the overall contrast. We attribute this to the decay of the longitudinal spin relaxation T 1 , which is observed as a consequence of the experimental method. The pulse sequence used in the experiment is shown in Supplementary  Figure 4a and consists of three laser pulses, each with a pulsewidth t 1 = 5 µs, that are repeated with a period t rep = 40 µs. To measure the contrast C = (I sig − I ref )/I ref , the photoluminescence is sampled during two windows synchronised with two of the laser pulses. The reference intensity I ref , is sampled following a long laser pulse and a short dark period t 0 . In this case the spins will be initialised to the bright, m s = 0 state. The signal, I M W is measured following a laser initialisation pulse, and a dark period that is dependent on the MW pulsewidth, τ . In the case of no microwave signal, sweeping τ measures the decay of T 1 as shown by the blue line in Supplementary Figure 4b. This decay is also present during conventional Rabi and CCD measurements, leading to the overall decay of contrast seen in the red and black lines of Supplementary Figure 4b, respectively.